inference by enumeration on Bayes graph

Question

a graph with known conditional prob between nodes, i.e. P(P2 | P1), P(P2 | ¬P1) are all known:

          P1
          ↓
          P2 
       ↓      ↓
      P3      P4

how can I use " inference by enumeration", to find P(P1 | ¬P3, p4)

Answer 1

$$\def\P{\operatorname{\sf P}}\begin{align}\P(P_1\mid \neg P_3,P_4) &=\dfrac{\displaystyle\P(P_1)\sum_{\small p_2\in\{P_2,\neg P_2\}}\P(p_2\mid P_1)\P(\neg P_3\mid p_2)\P(P_4\mid p_2)}{\displaystyle\sum_{\small p_1\in\{P_1,\neg P_1\}}\sum_{\small p_2\in\{P_2,\neg P_2\}}\P(p_1)\P(p_2\mid p_1)\P(\neg P_3\mid p_2)\P(P_4\mid p_2)}\end{align}$$